Finite Group Databasebeta

← #29663801

Subgroups — #29663801

Order 15616.

798 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456394C211yesmaximal, characteristic, complemented
2#21455285C211yesmaximal, characteristic, complemented
3#21455279C211yesmaximal, characteristic, complemented
4C2×#16839827C211yesmaximal, characteristic, complemented
5#21445259C212yesmaximal, complemented
6#21445255C212yesmaximal, complemented
7#21445207C212yesmaximal, complemented
8#21445173C212yesmaximal, complemented
9#21444577C211yesmaximal, characteristic, complemented
10#21444494C211yesmaximal, characteristic, complemented
11C61×#1547C211yesmaximal, characteristic, complemented
12C2×#14519805—21no—
13C2×#14519802—21no—
14C22×C8×D61—21no—
15#16840312C2211yescharacteristic
16C2×C4×D244C2211yescharacteristic
17#16840165C2211yescharacteristic, complemented
18#16840138C2211yescharacteristic, complemented
19#16840110C2211yescharacteristic, complemented
20#16840108C2211yescharacteristic, complemented
21#16839996—21no—
22#16839996—21no—
23#16839978—22no—
24#16839974—22no—
25#16839968C2212yes—
26#16839964C2212yes—
27#16839958C2212yes—
28#16839954C2212yes—
29C4×#11758199—21no—
30C4×#11758199—21no—
31#16839827—21no—
32#16839827—21no—
33#16839827C2212yes—
34#16839827C2212yes—
35#16839786C2211yescharacteristic
36#16839764—21no—
37#16839763—21no—
38C2×#14519689—21no—
39C2×C4×#92418C2211yescharacteristic
40#16839514C2212yescomplemented
41#16839514C2212yescomplemented
42#16839507C2212yescomplemented
43#16839505C2212yescomplemented
44C61×#532C2211yescharacteristic
45C61×#423C2211yescharacteristic
46C2×C4⋊C8×C61C2211yescharacteristic
47C61×#332C2212yescomplemented
48C61×#331C2212yescomplemented
49C2×#11758222—21no—
50C22×D244D411yescharacteristic
51C22×C4×D61—21no—
52#14519805—44no—
53#14519802—44no—
54C2×#11758190—22no—
55C2×#11758201—22no—
56C2×D488—22no—
57C2×#11758199—22no—
58C2×#11758199—21no—
59C2×#11758199—21no—
60C2×#11758194—21no—
61C22×#92418D411yescharacteristic
62C22×#92418D411yescharacteristic
63C2×C8×D61—22no—
64C2×C8×D61—22no—
65C2×C8×D61—44no—
66#14519777C2311yescharacteristic
67#14519774C2311yescharacteristic
68#14519765—21no—
69#14519751—21no—
70#14519751—21no—
71C4×D244—21no—
72C4×D244—21no—
73C4×D244—21no—
74C4×D244—21no—
75C4×D244D412yes—
76C4×D244C2312yes—
77C2×C4⋊Dic61D411yescharacteristic
78C2×#11758198—21no—
79#14519710—21no—
80#14519710—21no—
81C4⋊#92418—21no—
82C4⋊#92418C2312yes—
83#14519702—22no—
84#14519701—22no—
85#14519699—22no—
86#14519698—22no—
87#14519698—21no—
88#14519698—21no—
89#14519697—21no—
90#14519697—21no—
91#14519689—22no—
92#14519689—22no—
93C4×#92418D411yescharacteristic
94C4×#92418D411yescharacteristic
95C4×#92418C2311yescharacteristic
96C4×#92418C2311yescharacteristic
97C2×#41×C61—21no—
98C2×M16×C61—21no—
99C22×C8×C61—21no—
100C4⋊Q8×C61C2311yescharacteristic
101#127×C61C2311yescharacteristic
102#124×C61—21no—
103#121×C61—21no—
104#121×C61—21no—
105C4×D4×C61—21no—
106C4×D4×C61—21no—
107C2×C42×C61C2311yescharacteristic
108C4⋊C8×C61—21no—
109C4⋊C8×C61—21no—
110C4⋊C8×C61C2312yes—
111C4⋊C8×C61C2312yes—
112C23×D61—21no—
113#41×C61—42no—
114#41×C61—42no—
115C2×Q8×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61—41no—
120C22×C4×C61—41no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61#4111yescharacteristic
124C2×C22⋊D61—41no—
125C22×Dic61—21no—
126#11758222—42no—
127#11758222—42no—
128C2×D244—21no—
129C2×D244—22no—
130C2×D244—21no—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244D814yescomplemented
134C2×D244C2×D412yes—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—42no—
138C2×C4×D61—42no—
139C2×C4×D61—41no—
140C2×Dic122—21no—
141M16×C61—42no—
142C2×C8×C61—22no—
143C2×C8×C61—22no—
144C2×C8×C61—42no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C22⋊C4×C61—41no—
148C22⋊C4×C61—41no—
149C42×C61C2×D411yescharacteristic
150C42×C61C2×D411yescharacteristic
151C42×C61#4111yescharacteristic
152C42×C61C2411yescharacteristic
153#11758201—44no—
154#11758199—42no—
155#11758199—44no—
156#11758199—42no—
157#11758198—22no—
158#11758198—22no—
159#11758198—41no—
160#11758198—41no—
161C4⋊Dic61—21no—
162C4⋊Dic61—21no—
163C4⋊Dic61—21no—
164C4⋊Dic61C2×D412yes—
165#11758194—42no—
166C2×#92418—22no—
167C2×#92418—22no—
168C2×#92418—41no—
169C2×#92418C2×D411yescharacteristic
170C2×#92418C2×D411yescharacteristic
171C2×#92418C2×D411yescharacteristic
172C2×#92418C2×D411yescharacteristic
173D488—44no—
174#11758190—44no—
175C8×D61—44no—
176C8×D61—44no—
177C23×C61—41no—
178C23×C61#12111yescharacteristic
179C22×D61—22no—
180C22×D61—24no—
181C22×D61—42no—
182C22×D61—42no—
183C22×D61—41no—
184Q8×C61—41no—
185Q8×C61—41no—
186D4×C61—41no—
187D4×C61—41no—
188D4×C61—41no—
189D4×C61—41no—
190D4×C61—81no—
191D4×C61—82no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—82no—
197C2×C4×C61#12111yescharacteristic
198C2×C4×C61#12111yescharacteristic
199C2×C4×C61#12111yescharacteristic
200C2×C4×C61C2×D812yes—
201C2×C4×C61#13512yes—
202C2×C4×C61C22×D411yescharacteristic
203C2×C4×C61C22×D411yescharacteristic
204C2×C4×C61C2×#4111yescharacteristic
205C22⋊D61—82no—
206C2×Dic61—22no—
207C2×Dic61—42no—
208C2×Dic61—41no—
209D244—22no—
210D244—24no—
211D244—41no—
212D244—41no—
213D244—41no—
214D244—41no—
215D244—81no—
216D244C2×D814yes—
217C4×D61—44no—
218C4×D61—42no—
219C4×D61—42no—
220C4×D61—82no—
221Dic122—41no—
222Dic122—41no—
223C8×C61—42no—
224C8×C61—42no—
225#92418—21no—
226#92418—21no—
227#92418—21no—
228#92418—42no—
229#92418AGL(1,ℤ/8)12yescomplemented
230#19255—611nomaximal, Sylow
231C22×C61—21no—
232C22×C61—21no—
233C22×C61—81no—
234C22×C61—81no—
235C22×C61—81no—
236C22×C61#46612yes—
237C22×C61C2×#12111yescharacteristic
238C2×D61—24no—
239C2×D61—48no—
240C2×D61—42no—
241C2×D61—42no—
242C2×D61—81no—
243C2×D61—81no—
244C2×D61—81no—
245C4×C61—21no—
246C4×C61—21no—
247C4×C61—21no—
248C4×C61—81no—
249C4×C61#46612yes—
250C4×C61C22×D811yescharacteristic
251C4×C61C2×#13511yescharacteristic
252C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
253C4×C61#57511yescharacteristic
254Dic61—42no—
255Dic61—81no—
256#3169—611no—
257#3036—611no—
258#3036—611no—
259C2×C8×D4—611no—
260#1681—612no—
261#1681—612no—
262#1673—612no—
263#1667—612no—
264#1567—611no—
265#1547—611no—
266#1547D6111yescharacteristic, complemented
267C2×C61—22no—
268C2×C61—161no—
269C2×C61#303611yescharacteristic
270C2×C61C2×#46611yescharacteristic
271C2×C61#306711yescharacteristic
272D61—44no—
273D61—161no—
274C2×AGL(1,ℤ/8)—1221no—
275C2×AGL(1,ℤ/8)—1221no—
276C23×C8—1221no—
277#532—611no—
278#532C2×D6111yescharacteristic
279C2×C4×D4—611no—
280#500—611no—
281#493—611no—
282#486—611no—
283#486—611no—
284#466—1221no—
285#466—1221no—
286#463—1222no—
287#463—1222no—
288#461—612no—
289#461—612no—
290#459—612no—
291#459—612no—
292C4×D8—1221no—
293C4×D8—1221no—
294C8×D4—612no—
295C8×D4—612no—
296C8×D4—1221no—
297C8×D4—1221no—
298#423—611no—
299#423C2×D6111yescharacteristic
300C2×C4⋊C8C2×D6111yescharacteristic
301#417—1221no—
302#417—1221no—
303C2×C22⋊C8—1221no—
304C2×C4×C8—611no—
305#335—612no—
306#335—612no—
307#332—612no—
308#332C2×D6112yescomplemented
309#331—612no—
310#331C2×D6112yescomplemented
311C61#1925511yescharacteristic, complemented, Sylow
312C2×#41—21no—
313C2×#41—1221no—
314C22×D4—611no—
315C23×C4—1221no—
316AGL(1,ℤ/8)—2444no—
317AGL(1,ℤ/8)—2444no—
318C2×SD16—1222no—
319C2×SD16—1222no—
320C2×D8—1222no—
321C2×D8—1221no—
322C2×D8—1221no—
323C2×D8—1222no—
324C2×M16—21no—
325C2×M16—1221no—
326C22×C8—21no—
327C22×C8—611no—
328C22×C8—611no—
329C22×C8—1222no—
330C22×C8—1222no—
331C22×C8—2444no—
332C4⋊Q8—611no—
333C4⋊Q8C22×D6111yescharacteristic
334#127—611no—
335#127C22×D6111yescharacteristic
336#124—21no—
337#124—1221no—
338#121—21no—
339#121—21no—
340#121—1221no—
341#121—1221no—
342C4×D4—21no—
343C4×D4—21no—
344C4×D4—612no—
345C4×D4—612no—
346C4×D4—1221no—
347C4×D4—1221no—
348C4×D4—1221no—
349C4×D4—1221no—
350C2×C4⋊C4—611no—
351C2×C22⋊C4—1221no—
352C2×C42C22×D6111yescharacteristic
353#107—1221no—
354#107—1221no—
355C4⋊C8—21no—
356C4⋊C8—21no—
357C4⋊C8—612no—
358C4⋊C8—1221no—
359C4⋊C8C22×D6112yes—
360C4⋊C8C22×D6112yes—
361#103—1222no—
362#103—1222no—
363#102—1221no—
364#102—1221no—
365#102—1222no—
366#102—1222no—
367#102—1221no—
368#102—1221no—
369C22⋊C8—1222no—
370C22⋊C8—1222no—
371C4×C8—611no—
372C4×C8—611no—
373C4×C8—611no—
374C4×C8—611no—
375C24—1221no—
376#41—42no—
377#41—42no—
378#41—2442no—
379#41—2442no—
380C2×Q8—21no—
381C2×Q8—1221no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—41no—
386C2×D4—612no—
387C2×D4—614no—
388C2×D4—1221no—
389C2×D4—1221no—
390C2×D4—1221no—
391C2×D4—1221no—
392C2×D4—1222no—
393C2×D4—2441no—
394C22×C4—41no—
395C22×C4—1221no—
396C22×C4—1222no—
397C22×C4—1222no—
398C22×C4—2442no—
399C22×C4—2442no—
400C22×C4—2441no—
401C22×C4#1175822211yescharacteristic
402C22×C4D4×D6111yescharacteristic
403C22×C4D4×D6111yescharacteristic
404SD16—2444no—
405SD16—2444no—
406D8—2444no—
407D8—2442no—
408D8—2442no—
409D8—2444no—
410M16—42no—
411M16—2442no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—42no—
415C2×C8—611no—
416C2×C8—611no—
417C2×C8—611no—
418C2×C8—611no—
419C2×C8—1222no—
420C2×C8—1222no—
421C2×C8—2444no—
422C2×C8—2444no—
423C2×C8—2441no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—612no—
427C4⋊C4—1221no—
428C4⋊C4—1221no—
429C4⋊C4—1221no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—1222no—
433C22⋊C4—1222no—
434C22⋊C4—2441no—
435C22⋊C4—2441no—
436C42#1175822211yescharacteristic
437C42D4×D6111yescharacteristic
438C42D4×D6111yescharacteristic
439C42C23×D6111yescharacteristic
440C23—41no—
441C23—1224no—
442C23—1222no—
443C23—2442no—
444C23—2442no—
445C23—2441no—
446C23#1451975011yescharacteristic
447Q8—41no—
448Q8—41no—
449Q8—2441no—
450Q8—2441no—
451D4—41no—
452D4—41no—
453D4—41no—
454D4—41no—
455D4—81no—
456D4—82no—
457D4—614no—
458D4—1224no—
459D4—1222no—
460D4—4881no—
461D4—4882no—
462D4—2441no—
463D4—2441no—
464D4—2441no—
465D4—2441no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—82no—
471C2×C4—1222no—
472C2×C4—4882no—
473C2×C4—2444no—
474C2×C4—2441no—
475C2×C4—2442no—
476C2×C4—2442no—
477C2×C4—2442no—
478C2×C4#1451975011yescharacteristic
479C2×C4#1451975011yescharacteristic
480C2×C4#1451975011yescharacteristic
481C2×C4D8×D6112yes—
482C2×C4#1451979812yes—
483C2×C4C2×D4×D6111yescharacteristic
484C2×C4C2×D4×D6111yescharacteristic
485C2×C4C2×#1175822211yescharacteristic
486C8—42no—
487C8—42no—
488C8—612no—
489C8—1221no—
490C8—1221no—
491C8—1221no—
492C8—2442no—
493C22—21no—
494C22—21no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22—1224no—
499C22—4881no—
500C22—4881no—
501C22—4881no—
502C22—2448no—
503C22—2442no—
504C22—2442no—
505C22#1683999712yes—
506C22C2×#1451975011yescharacteristic
507C4—21no—
508C4—21no—
509C4—21no—
510C4—81no—
511C4—4881no—
512C4—2442no—
513C4#1683999712yes—
514C4C2×D8×D6111yescharacteristic
515C4C2×#1451979811yescharacteristic
516C4AGL(1,ℤ/8)×D6111yescharacteristic
517C4#1684056111yescharacteristic
518C2—22no—
519C2—161no—
520C2—9761no—
521C2—2444no—
522C2#2145528211yescharacteristic
523C2C2×#1683999711yescharacteristic
524C2#2145553811yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1547C2
socleC22×C61C2×#121
hypercenter#1547D61
hypocenterC61#19255
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1547

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #21456394C2semidirect
2. #21455285C2semidirect
3. #21455279C2semidirect
4. C2×#16839827C2semidirect
5. #21445259C2semidirect
6. #21445255C2semidirect
7. #21445207C2semidirect
8. #21445173C2semidirect
9. #21444577C2semidirect
10. #21444494C2semidirect
11. C61×#1547C2semidirect
17. #16840165C22semidirect
18. #16840138C22semidirect
19. #16840110C22semidirect
20. #16840108C22semidirect
40. #16839514C22semidirect
41. #16839514C22semidirect
42. #16839507C22semidirect
43. #16839505C22semidirect
47. C61×#332C22semidirect
48. C61×#331C22semidirect
133. C2×D244D8semidirect
229. #92418AGL(1,ℤ/8)semidirect
266. #1547D61semidirect
308. #332C2×D61semidirect
310. #331C2×D61semidirect
311. C61#19255semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214563942
#214552852
#214552792
C2×#168398272
#214452592
#214452552
#214452072
#214451732
#214445772
#214444942
C61×#15472
#1925561